Crystal families and systems in higher dimensions, and geometrical symbols of their point groups. I. Crystal families in five-dimensional space with two-, three-, four- and sixfold symmetries.
The aim of this paper and of the following one [Weigel, Phan & Veysseyre (2008). Acta Cryst. A64, 687-697] is to complete the list of the Weigel-Phan-Veysseyre (WPV) symbols of the point groups of space E5 that was started in previous papers and in two reports of an IUCr Subcommittee on the Nomenclature of n-Dimensional Crystallography. In this paper, some crystal families of space E5 are studied. The cells of these are right hyperprisms with as a basis either two squares, or two hexagons, or a square and a hexagon. If the basis is made up of two squares, the two families are the (monoclinic di squares)-al family (No. XVI) and the (di squares)-al family (No. XIX). If the basis is made up of two hexagons, the two families are the (monoclinic di hexagons)-al family (No. XVII) and the (di hexagons)-al family (No. XXI). If the basis is made up of one square and one hexagon, the family is the (square hexagon)-al family (No. XX). In order to link space E5 to spaces E2, E3 and E4, some results published in previous papers are recalled. In fact, most of the symbols of the point groups of space E5 can be deduced from the symbols of the four, six and 23 crystal families of spaces E2, E3 and E4, respectively.
Veysseyre
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R
Laboratoire Mathématiques Appliquées aux Systèmes, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry Cedex, France. renee.veysseyre@normalesup.org
Weigel
D
D
Phan
Th
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